1 Introduction

Proportional reasoning abilities are seen as important skills to thrive in our modern world (Lesh et al., 1988). Take, for example, the present COVID-19 crisis, because of which we are bombarded with proportional information. We are informed about the proportion of the population that has received a first dose of the vaccination, or about the probability of catching COVID-19 when we are not wearing a mask. Traditionally, proportions are considered to be concepts too complex for children under the age of 11 to fully grasp (Inhelder & Piaget, 1958) and, as a consequence, they are often introduced rather late in the elementary school mathematics curriculum (Van Dooren et al., 2018). This late introduction is remarkable considering that more recent research showed that young children are already able to understand proportional information to a certain extent (Boyer & Levine, 2012; Vanluydt et al., 2022) and that this understanding can be stimulated at an early age (Boyer & Levine, 2015). Piaget and Inhelder (1951/1975) not only elaborately investigated young children’s proportional reasoning abilities, but also focused on children’s probabilistic reasoning abilities. They concluded that, similar to the understanding of proportions, fully grasping the concept of probability requires formal operational thinking, which is generally not present before the age of 11. Bryant and Nunes (2012) found that even after the formal operational stage is reached, many children and even adults still tend to struggle with probability. Often, erroneous probabilistic reasoning is attributed to an insufficient understanding of proportionality (Begolli et al., 2021; Bryant & Nunes, 2012).

Luckily, albeit late, once proportions are introduced in the curriculum, proportional reasoning receives ample attention in the elementary school mathematics class. Children get extensive practice solving problems referring to numerous proportional contexts, such as fair-sharing, buying a number of products for a certain price, distance travelled in a certain time, recipes, and mixtures. One of these contexts, the context of probability, is often neglected in elementary school (Van Dooren et al., 2006). For example, in Flanders, Belgium, the National Educational Standards for elementary school “probability” is mentioned only once and it reads “Recognizing through examples that fractions can be understood as a part of a ratio, a distribution, a division, a multiplier, a number or the representation of a probability”. Langrall (2018) showed that the topic of probability is also lacking in the elementary school curriculum of many other countries, e.g., Germany and the UK. Would the underlying idea of educators and textbook developers be that probabilistic reasoning is also indirectly stimulated when other proportional contexts are introduced? Nonetheless, Falk et al. (2012) gave a fair warning that proportional and probabilistic reasoning are not interchangeable and that probability should not be cast aside for elementary school children. They pointed out that probabilities always involve an element of uncertainty, whereas this is not the case in mere proportional contexts. However, research has repeatedly used probabilistic tasks to make general claims about children’s proportional reasoning, without taking into account that, perhaps, findings based on tasks involving an element of uncertainty cannot be generalized to other proportional contexts (e.g., Chapman, 1975; Hurst & Cordes, 2018; Jeong et al., 2007; Ruggeri et al., 2018; Singer & Resnick, 1992). Indeed, findings on children’s proportional reasoning abilities vary strongly across studies (e.g., Hurst & Cordes, 2018; Vanluydt et al., 2020). Could it be that these differences in findings and conclusions are explained by the fact that studies use different proportional contexts to measure proportional reasoning?

In the present study, we argue that the context of proportional problems not only has an impact on the difficulty of the problem, but that it also plays an important role in how children approach the problems. In other words, different contexts might elicit different (erroneous) thinking strategies. In short, the aim of our study is to investigate the role context plays in young children’s proportional reasoning. To this end, we will, first, define proportional reasoning and give a short overview of the tasks and contexts that have been used in research on proportional reasoning, followed by discussion of several similarities and differences between those contexts.

1.1 Proportional reasoning in young children

Proportions are characterized by the equality of two ratios (mathematically written as a/b = c/d). Proportional reasoning refers to the ability to understand, construct, and use the multiplicative relationship between two co-varying measures (Resnick & Singer, 1993; Vergnaud, 1988). A typical example where proportional reasoning is needed in daily life is adjusting recipes according to the number of servings needed: If you need two eggs in a recipe for four people, you will need three eggs in a recipe for six people (2/4 = 3/6). Besides its importance in daily life, there is unanimity about the important role of proportional reasoning throughout a learner’s mathematical development. Proportional reasoning has been described as a watershed concept, the capstone of elementary concepts, and a cornerstone of higher mathematics (Lesh et al., 1988). Even though learning to reason proportionally receives a lot of instructional attention in elementary education—typically from the fourth grade on—it remains to be a stumbling block (Resnick & Singer, 1993). Many adolescents and even adults seem to be particularly bad at it (e.g., Lovell, 1961; Noelting, 1980).

The late introduction of proportional reasoning in the elementary curriculum might stem from the longstanding line in research that claims that children can only start to reason proportionally when they have reached the formal-operational level of cognitive thinking, generally not obtained before the age of 11 (Inhelder & Piaget, 1958). It is argued that before that, children tend to quantify relations in an additive way, and only later, when they have reached the formal-operational level, can they quantify these relations in a multiplicative way, and thus understand proportions (Inhelder & Piaget, 1958). A convincing body of studies supports the stance of Piaget and colleagues that additive reasoning is a precursor of proportional reasoning, that proportional reasoning is a rather late achievement, and that adolescents, and even adults, still struggle with it (e.g., Fujimura, 2001; Kaput & West, 1994; Karplus, 1981; Karplus et al., 1983; Lesh et al., 1988; Lovell, 1961; Noelting, 1980; Vergnaud, 1983, 1988).

More and more studies, however, have shown a renewed interest in the development of proportional reasoning in early childhood and indicate that children can reason proportionally in certain tasks before the formal-operational stage (e.g., Boyer & Levine, 2012; He et al., 2018; Hurst & Cordes, 2018; Vanluydt et al., 2022). Two major categories of proportional problems are used in research on proportional reasoning: comparison problems and missing-value problems. Most studies have investigated proportional reasoning using comparison problems (Boyer & Levine, 2012, 2015; Boyer et al., 2008; Möhring et al., 2015). In these problems, children are required to make a judgement about the values of given ratios in terms of whether or not they are equal, or which is greater (Cramer et al., 1993; Harel & Behr, 1989; Tourniaire & Pulos, 1985). A common context for proportional comparison problems are mixtures. For example, Boyer and colleagues (2008) provided 6- to 9-year-olds with visuals of mixtures of juice and water. Out of two given mixtures, they had to identify the one that contained the same ratio of water and juice (and thus the same taste) as the target ratio in a given third mixture. They found that 6-year-olds were able to, on average, solve about half of the trials and the number of trials that were solved increased with age. Nine-year-olds were able to solve, on average, 80% of trials. However, children experienced greater difficulties with proportional reasoning when these trials involved discrete units (e.g., separate cups of juice and water) compared to problems with continuous quantities of juice and water (e.g., heights of liquid in a cylinder). The authors suggest that children’s difficulties with discrete quantities are due to children’s propensity to match quantities numerically instead of proportionally.

Many other studies using proportional comparison problems involve probability contexts (Falk et al., 2012). For example, Hurst and Cordes (2018) investigated 3- to 6-year-olds’ proportional reasoning abilities using the context of two given spinners, where children had to identify the one that would help them win more stickers. Remarkably, they found that, in contrast to findings of Boyer and colleagues (2008), 5- to 6-year-olds already performed above chance level on continuous trials (75% correct) and also on discrete trials (80% correct). Ruggeri et al., (2018, p. 6) also investigated children’s proportional reasoning performance using a probabilistic context. In their study, 7- and 10-year-olds, as well as adults, were shown two bags of gold and black tokens and were asked to identify the bag from which it was more likely to randomly draw a gold token. Their findings differed from Hurst and Cordes (2018), with 7-year-olds solving 61%, 9-year-olds solving 65%, and adults solving 84% of trials on average. Other studies using probability comparison tasks add to the confusion about children’s proportional reasoning abilities. Some studies suggest that preschoolers and infants are already able to succeed (e.g., Denison & Xu, 2014; Falk et al., 1980; Girotto et al., 2016) while other studies show that elementary school children still struggle (e.g., Falk et al., 2012; Piaget & Inhelder, 1958/1975). Canizares et al. (1997) found that even older children who were acquainted with fractions still struggled when comparing two probabilities.

Comparison problems are considered to elicit more intuitive responses, as they do not imply that children have to understand the mathematical symbols of proportions nor do they require them to make exact calculations (Matthews & Chesney, 2015). Erroneous strategies as well as guessing often lead to the correct response. Thus, such tasks require high numbers of trials in a balanced task design, taking into account several item characteristics (Falk et al., 2012). Falk and Wilkening (1998) pointed out that missing-value problems require fewer trials to investigate children’s ability to use the idea of proportionality, which is why the current study uses missing-value problems.

1.1.1 Proportional reasoning in missing-value problems

In a typical missing-value problem, three numbers, a, b, and c, are presented, and the task is to find the unknown x under a constraint of equality of the two ratios, such that a/b = c/x (Cramer et al., 1993; Harel & Behr, 1989; Tourniaire & Pulos, 1985). For instance: How many berries do you need for 3 glasses of juice, if you need 10 berries for 6 glasses (6/10 = 3/x, x = ?). In this type of task, it is harder to respond intuitively as children have to construct/calculate the exact numerosity for the missing quantity, even when such problems are presented in a non-symbolic way (Falk & Wilkening, 1998). Fewer studies have investigated proportional reasoning abilities in early childhood using missing-value problems. Resnick and Singer (1993) used a fair-sharing context, asking 5- to 7-year-old children to feed fish of different lengths in a missing-value task. They found that children tended to give larger amounts of food to larger fish, thereby solving the task in a proportional way. Children also tended to perform better when the nature of the quantities in the problem was discrete (i.e., countable; e.g., beads) than when the nature of the quantities was continuous (i.e., uncountable; e.g., strips of ribbon). Vanluydt et al. (2022) used a fair-sharing, missing-value task in which 5- to 8-year-old children had to equally distribute food (grapes or bars of chocolate) among puppets to assess their proportional reasoning abilities. They observed a gradual improvement from age 5 to 8 in children’s proportional reasoning abilities, but this improvement occurred slower for proportional reasoning with a discrete and continuous quantity than for proportional reasoning with two discrete quantities. Based on traditional research on proportional reasoning, they expected to observe a lot of erroneous additive reasoning as a strategy to solve a different missing-value task; however, the opposite was true. Children used correct strategies such as equal distribution and repeated addition to come to the right solution. This was quite a surprise, especially given the results of previous studies using comparison problems in mixture or probability contexts (Boyer et al., 2008; Hurst & Cordes, 2018; Ruggeri et al., 2018) that had shown that children had difficulty with problems with discrete quantities, stemming from a propensity to compare quantities on the basis of the number of elements in the target quantity (i.e., an additive comparison), rather than on the basis of proportional relations (i.e., a multiplicative comparison).

However, missing-value tasks presented in probabilistic contexts paint a different picture than missing-value problems using a mere proportional fair-sharing context. In a missing-value probability task, children receive two sets. One set is complete, containing elements of two types (a winning and losing color). The other set is incomplete, containing only elements of the losing color. Children are required to complete the incomplete set by adding elements of the winning color in such a way that the probability of drawing an element of the winning color is equal in both sets (Falk & Wilkening, 1998). Falk and Wilkening (1998) found that young children primarily use a one-dimensional strategy to solve these missing-value problems: Children only attend to the numerator, adding the same number of winning color elements to the incomplete set as present in the complete set, disregarding the number of losing elements. Only from age nine do children start to attend to both the number of winning and losing elements in these missing-value problems, although, generally combining them in erroneous additive ways. From age 11, children start to correctly apply proportionality to solve probabilistic missing-value problems; however, only by age 13 do children really thrive at it.

These findings are in sharp contrast with children’s performance on missing-value problems with a proportional fair-sharing context, mentioned above. In the following paragraph, we zoom in on similarities and differences between the different contexts used in proportional reasoning tasks as a possible explanation for varying findings.

1.1.2 Contexts for proportional reasoning

As presented above, some commonly used contexts in research on children’s probabilistic reasoning are as follows: fair-sharing (e.g., sharing grapes among children), mixtures (e.g., making juices from water and orange concentrate), and probability (e.g., spinner comparisons). Fair-sharing is considered a very familiar context for young children (Schmidt & Sommerville, 2011) and because of this familiarity, children might be better at applying proportional reasoning in these contexts compared to mixture and probability contexts.

However, besides familiarity, less obvious aspects related to contexts might affect children’s proportional reasoning. A first aspect is that there can be a difference in the degree of physical similarity between the measure spaces in the different contexts. In the fair-sharing context (e.g., a number of grapes assigned to a number of puppets), the two measure spaces are usually more dissimilar compared to the mixtures or probability context (e.g., a number of red and blue berries in an urn). A study of Lawton (1993) has shown that university students perform better on proportional reasoning problems with less similar measure spaces compared to problems with more similar measure spaces. It seemed that when the distinction between the two measure spaces was more clear, students were more prone to approaching the problems proportionally.

A second aspect that might affect children’s proportional reasoning is the presence of uncertainty. As previously mentioned, it has been criticized to throw proportional and probabilistic reasoning in one pot, because with probability, an element of uncertainty (i.e., the correct choice will not guarantee success) comes into play. Assuming all apples of the same color are equal in size and taste, and you have two apples for one child, you will need four apples for two children, for all children to have an equal number of apples. Or when in a mixture context two sour green apples and one sweet red apple are blended together in a jar to make apple juice, you will need four green apples to blend with two red apples if you want the juice in the second jar to taste exactly the same as the first apple juice. Thus, in fair-sharing and mixture contexts, the correct answer and the outcome correspond, with certainty. However, in probabilistic problems, the correct answer and the eventual outcome might contradict each other (Falk et al., 2012): Imagine you got a bag with one red and two green apples, and another bag with two red and three green apples. Drawing an apple from the second bag gives you a larger probability of drawing a red apple, but when you actually draw an apple from each bag, it may happen that you draw a red apple from the first bag and a green one from the second bag. Probability involves an element of uncertainty and thus the correct choice will not always be followed by the desired/intended outcome (Falk & Konold, 1992). Falk et al. (2012) suggested that some children might be arithmetically proficient in calculating proportions, but hold false beliefs when faced with uncertainty. For example, they might believe that the bag with the most green apples results in a higher probability because green is a lucky color; therefore, the number of green apples has to be the same to get the same probability. Moreover, the equiprobability bias might cause children to believe that the number of apples does not matter at all, because each color has a 50–50% probability of being drawn, independent from the number of apples of each color in each bag (Amir & Williams, 1999; Green, 1982). Another phenomenon that has been frequently observed in adults when they are reasoning in uncertain situations is called denominator neglect (e.g., Denes-Raj et al., 1995; Garcia-Retamero et al., 2010). When you forget about the importance of the denominator, you will, for example, pay too much attention to the events that you want/want not to happen and forget about all the events that might happen. Thus, for an equal probability of drawing a red apple, neglecting the denominator would mean you would think the number of red apples has to be the same in both bags, without taking into account the number of green apples. This phenomenon has also been observed in children (e.g., Acredolo et al., 1989; Falk et al., 2012). Taken together, one might succeed in proportional reasoning tasks, but flunk probabilistic reasoning tasks because of false beliefs and/or not knowing/forgetting that the probability of drawing a green apple is determined by the proportions of green and red apples within the bags.

A third aspect that might affect children’s proportional reasoning concerns the fact that a probability is not tangible or directly observable. This refers to the difference between intensive and extensive quantities. An extensive quantity is an absolute quantity that can be expressed as the sum of its parts and represented in one single value (e.g., length). An intensive quantity is a ratio between two extensive quantities and does not depend on the amount of extensive quantity involved as long as the ratio remains constant (e.g., speed) (Howe et al., 2010; Kaput & West, 1994). In a fair-sharing context, there is no need to reason about intensive quantities and problems can be solved by direct modelling (i.e., equal distribution or repeated addition). Fair-sharing results in an extensive quantity that is directly observable: Each person gets a certain number of apples. Mixtures, however, concern an intensive quantity (e.g., flavor). It can be experienced directly, though in a less evident way: You cannot see the flavor, but you can still taste it. Probability also concerns an intensive quantity. It is determined by a number of tangible elements (e.g., red and green apples), but the probability of an event (e.g., blindly drawing a green apple) is not tangible as it cannot be observed. You cannot directly see a probability in the occurrence of one event, and at best one can estimate it from a long series of trials.

1.2 The present study

In the present study, we will use different missing-value contexts to investigate the proportional reasoning abilities of young elementary school children before receiving elaborate formal instruction in it. More specifically, we aim to investigate whether differences exist in children’s performance on missing-value problems when arithmetic requirements are identical, but different proportional and probabilistic contexts are used. Based on previous studies (Falk & Wilkening, 1998; Supply et al., 2020a, b; Vanluydt et al., 2022), we would also expect the fair-sharing context to be easier for children compared to the probability context. Falk et al. (2012) moreover suggested that, compared to mere proportional situations, children might be more susceptible to holding false beliefs in probabilistic situations, which can result in erroneous responses. Therefore, besides taking into account children’s correct responses, we aim to investigate whether children’s tendency to give responses that would be predicted by popular erroneous strategies differs across contexts. More concretely, we will focus on one-dimensional answers (i.e., only attending to the numerator) and additive answers (i.e., using the absolute difference between the terms of the ratio).

2 Method

2.1 Participants and procedure

The present study is part of a large-scale longitudinal study on the development of young children’s early mathematical competencies (www.ppw.kuleuven.be/o_en_o/Wisenco). Seventeen schools were selected based on governmental data to obtain a diverse and representative sample in terms of socioeconomic status (SES). All children attending second year of preschool in the school year 2016–2017 were invited to participate, meaning they would be followed until third year of elementary school. An active informed consent was received for 410 children. For the present study, data of the final time point was used, when children were in third year of elementary school (spring 2021). In Belgium, children enroll in six years of elementary school starting from the ages of 5 to 6. At the moment of testing in third year of elementary school, children had a mean age of 8 years and 10 months (SD = 3.47 months, range 8 years 4 months–9 years 4 months). Due to switching to a different school, several children left the sample over the course of four years. Complete data were available for 305 participants (153 girls). The children in our sample had not yet received any formal instruction in proportional or probabilistic reasoning.

2.2 Materials

Children received a test-battery consisting of three individual tasks. Together, these tasks took about 30 minutes to complete. Each of the tasks consisted of six items. All items were missing-value problems, in which children had to construct a set B equivalent to a set A by putting the elements in set B in the same ratio as the elements in set A. Set A contained two types of elements, namely white and black berries or puppets and black berries. Set B contained only one type of element, namely white berries or puppets and had to be completed by adding black berries. For each item, the number of black berries that would be predicted by correct proportional thinking or by the erroneous thinking strategies, namely, one-dimensional or additive strategies, differed. When children responded with a number of berries different from the number predicted by one of the strategies, the response was coded as “other.” Table 1 shows the number of elements that were repeatedly used in each task, as well as the correct response and the responses that would be predicted by one-dimensional or additive thinking. The tasks are presented below in the set order of administration. The process of task administration was identical for all three tasks.

Table 1 The number of elements used in each task and the correct responses, as well as the responses predicted by erroneous one-dimensional (OD) or additive (ADD) thinking. Set A contained elements of two types (puppets or white berries and black berries). Set B contained elements of one type (puppets or white berries) and had to be completed by adding black berries
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2.2.1 Probability context

The missing-value task of Supply et al. (2020a, b) was used. Children were introduced to two identical birds presented on animated slides. The birds love black berries and hate white berries. The task consisted of six items. For every item, each bird received a box containing a different number of berries. Children were explained that each trial, each bird could blindly draw one berry from its own box. The box of the first bird always contained black and white berries, but the box of the second bird only contained white (undesirable) berries. Children had received a booklet in which the same items were visually presented. They were asked to add blackberries to the box of the second bird to make it a fair game by drawing black berries in the box of the second bird in their booklet (see Fig. 1). They were repeatedly instructed that they could not colorize the given white berries, but that they had to draw additional black berries. The test administrator intervened when children did colorize white berries. The internal consistency for the complete item set was good (α = 0.85).

Fig. 1
figure 1

Example item for the creating equal probabilities task

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2.2.2 Mixture context

We adjusted the context of the missing-value task of Supply et al. (2020a, b) to create a missing-value proportional reasoning task in a mixing context. Children were introduced to a new bird that loves making juice of black and white berries, but always wants the juice in two mixers to taste exactly the same. The bird already added white and black berries in the first mixer, but only white berries in the second mixer. Children had received a booklet in which the items were visually presented, with exactly the same numerical properties (i.e., ratios) as in the other two tasks. In each trial, the task was to draw the black berries in the second mixer so the juices would taste exactly the same (see Fig. 2). Again, children were told not to colorize already present white berries. The internal consistency for the complete item set was good (α = 0.89).

Fig. 2
figure 2

Example item for the creating equal mixtures task

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2.2.3 Fair-sharing context

The fair-sharing missing-value task of Supply et al. (2020a, b) was adjusted to be comparable to the mixing and probability task. Again, children received a booklet in which the items were visually presented, with exactly the same numerical properties (i.e., ratios) as in the other two tasks. In each of the trials, two situations were visually presented, in which one or more birds received a number of berries. In the first situation, the birds and the number of berries were given, and in the second situation, the children had to draw the berries that the birds received so that it would be fair (see Fig. 3). The internal consistency for the complete item set was good (α = 0.79).

Fig. 3
figure 3

Example item for the fair-sharing task

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2.3 Statistical analysis

All data were analyzed using statistical software packages SPSS and JASP. Descriptive statistics were calculated to compare the scores on the three reasoning tasks. Violin plots were used to visualize children’s performance across tasks. To test our primary hypothesis that the context in which missing-value problems are presented affects children’s performance, a repeated measures ANOVA with one within factor (task) was conducted with the overall number of correct, one-dimensional, or additive responses on the tasks as the dependent variable. To account for violations of the assumption of sphericity, degrees of freedom were adjusted according to the Huynh–Feldt correction. Fisher’s LSD procedure was used for post-hoc pairwise comparisons (Levin et al., 1994). General η2 and Cohen’s d were calculated to assess the size of the effects (Lakens, 2013).

3 Results

Descriptives and correlations between each of the tasks and the total scores are found in Table 2. Violin plots of the total number of correct, one-dimensional, and additive responses on each of the tasks are shown in Figs. 4, 5, and 6.

Table 2 Means, standard deviations, and correlations for the total number of correct, one-dimensional (OD), and additive (ADD) responses on each of the tasks
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Fig. 4
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Violin plots of total correct responses on each of the tasks

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Fig. 5
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Violin plots of total one-dimensional responses on each of the tasks

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Fig. 6
figure 6

Violin plots of total additive responses on each of the tasks

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The violin plots suggest large differences within children over the three contexts. Children perform generally better on the fair-sharing task compared to the mixture and probability task. Differences between the mixture and probability task are less pronounced. Our main hypothesis that the context of the missing-value problems affects children’s overall performance was further confirmed by a large and statistically significant main effect, F(1.73, 525.01) = 867.7, p < 0.001, η2G = 0.53. Pairwise comparisons following Fisher’s LSD procedure showed that children solved significantly more items in the fair-sharing context correctly compared to the mixture context, t(304) = 32.28, p < 0.001, Cohen’s d = 1.85, and probability context, t(304) = 33.61, p < 0.001, Cohen’s d = 1.93. Children’s mean performance in the mixture context was exactly the same as their mean performance in the probabilistic context, t(304) = 0.00, p = 1, Cohen’s d = 0.00.

The violin plots, furthermore, suggest that erroneous responses as predicted by one-dimensional or additive thinking are rarely given in the fair-sharing context, but are sometimes more or less consistently given in the mixture and probability context. Regarding the one-dimensional responses, a large significant main effect for context was found, F(1.93, 585.48) = 142.2, p < 0.001, η2G = 0.17. Children gave significantly fewer one-dimensional responses in the fair-sharing context compared to the mixture context, t(304) =  − 11.20, p < 0.001, Cohen’s d =  − 0.64, and probability context, t(304) =  − 15.14, p < 0.001, Cohen’s d =  − 0.87. Moreover, a smaller but still significant difference was found between the mixture and the probabilistic context, t(304) =  − 6.19, p. < 0.001, Cohen’s d =  − 0.35: Children made significantly fewer one-dimensional errors in the mixture context than in the probabilistic context.

Regarding the additive responses, a large significant main effect for context was also found, F(1.98, 593.88) = 126.58, p < 0.001, η2G = 0.18. Children gave significantly fewer additive responses in the fair-sharing context compared to the mixture context, t(304) =  − 14.38, p < 0.001, Cohen’s d =  − 0.82, and probability context, t(304) =  − 12,45, p < 0.001, Cohen’s d =  − 0.71. Moreover, a smaller but still significant difference was found between the mixture and the probabilistic context, t(304) = 3.88, p. < 0.001, Cohen’s d = 0.22: Children gave significantly more additive responses in the mixture context than in the probabilistic context.

4 Discussion

In the present study, we aimed to investigate children’s proportional reasoning abilities using missing-value tasks in three different contexts that have been repeatedly used in literature (Boyer et al., 2008; Falk & Wilkening, 1998; Supply et al., 2020a, b; Vanluydt et al., 2022). These contexts entailed fair-sharing, mixture, and probability contexts. Presenting three contexts to the same sample of children allowed us to draw a more nuanced picture on children’s abilities to reason about proportions in missing-value problems compared to previous studies.

Traditionally, proportions are believed to be tough for young children to grasp and are introduced rather late in the elementary school curriculum (Inhelder & Piaget, 1958; Piaget & Inhelder, 1951/1975). Already in the 1950s, Piaget and Inhelder (1951/1975) concluded that formal operational thinking is needed to successfully reason about proportions and probability and this is generally not acquired before the age of 11. Despite these former claims, more and more research suggested that children are able to reason about probabilities and proportions before the formal operational stage (e.g., Bryant & Nunes, 2012; Vanluydt et al., 2022). For example, Vanluydt et al. (2022) showed that children are able to reason proportionally in missing-value problems presented in a fair-sharing context long before the formal operational age. These findings were confirmed in the present study with 8- to 9-year-old children showing strong proportional reasoning abilities to solve missing-value problems when a fair-sharing context was used.

However, our study also indicated that it would be too hasty to conclude that young children fully grasp proportionality before they have reached the formal operational, by merely looking at their performance in fair-sharing contexts. Indeed, when the same missing-value problems are presented in a different context, in this case a mixture or probabilistic context, children’s performance significantly deteriorates. Moreover, responses corresponding to erroneous rules, such as only taking into account the numerator or equalizing the absolute difference between the terms of the ratio, are more prevalent in the mixture and probabilistic contexts compared to the fair-sharing context. Importantly, the kind of erroneous response differed with the context: In the mixture context, we observed more erroneous additive responses, while the probabilistic context evoked more erroneous one-dimensional responses. Regarding the probabilistic context, these findings are in line with Falk and Wilkening (1998), who found that children struggled with applying proportionality to solve probabilistic missing-value problems until the age of 13. It has been argued that probabilistic situations might be more difficult for children than mere proportional situations due to the fact that probability entails uncertainty (Falk et al., 2012). This uncertainty might foster false beliefs that affect children’s tendency to use correct proportional reasoning. Our findings, indeed, suggest that a probabilistic context is more difficult for children than a mere proportional missing-value context that has been used before in research, namely, fair-sharing. However, if the proportional questions are framed in a mixture context, then the difference in the children’s average performance between the probabilistic and the proportional reasoning context disappears. Furthermore, another intriguing finding was that there is also a difference in the kind of errors that were evoked by the three contexts.

A possible explanation for these differences might be that children are more familiarized with a fair-sharing context compared to mixture and probability contexts. Previous research has shown that already before the age of two, young children understand and expect equal distributions (Schmidt & Sommerville, 2011). As already stated before, the mixture and the probability context are assumed to be more difficult to interpret for children, because they require children to have knowledge of more difficult principles on top of the scheme of proportionality. Other explanations might be at play too, especially regarding the large gap in performance between the fair-sharing context and the other two contexts.

Presenting these missing-value problems in different contexts led to two confounding variables that could have influenced children’s performance. First, there was a difference in the degree of physical similarity between the measure spaces in the different contexts. In the fair-sharing context, the two measure spaces (puppets and berries) were dissimilar compared to the other two contexts (black and white berries). Our results are in line with the findings of Lawton (1993), that students perform better when the distinction between the two measure spaces is more clear, but in a much younger age group. The children in our study performed better in the fair-sharing context, with more dissimilar measure spaces, compared to the other two contexts, with more similar measure spaces. Moreover, the contexts with similar measure spaces elicited more responses corresponding to the erroneous rules only taking into account the numerator or equalizing the absolute difference between the terms of the ratio. This raises the question if a different kind of reasoning is elicited when there is a difference in the physical similarity between the measure spaces. Perhaps, contexts with similar measure spaces such as the mixing and probabilistic contexts more easily elicit part-whole reasoning: One puts the black berries (part) in relation to the black and white berries (whole). The fair-sharing context with dissimilar measure spaces might facilitate part-part reasoning: One puts the black berries (part 1) in relation to the puppets (part 2). It has been suggested by Nunes and Bryant (1996) that it is easier for children to grasp proportions as part-part relations compared to fractions or part-whole relations. The same might be true for probabilities, but future studies are needed to confirm this hypothesis (Bryant & Nunes, 2012).

A second aspect that could have played a role in the differences in performance in the three contexts is the difference in reasoning about intensive and extensive quantities. The taste of a mixture is an intensive quantity: The taste (or sweetness) of the mixture depends on the ratio between the black and white berries. Similarly, the probability is an intensive quantity: It is not a larger number of elements that increases the probability of winning, it depends on the ratio between winning and losing elements. In a fair-sharing context, however, there is no need to reason about intensive quantities and problems can be solved by direct modeling (i.e., equal distribution or repeated addition). Since the early elementary school curriculum mainly focuses on extensive quantities, it does not come as a surprise that reasoning about intensive quantities is more difficult for young children (Nunes et al., 2003). Our study confirmed this by showing that the two contexts with intensive quantities were more difficult for 8- to 9-year-olds.

4.1 Strengths and limitations

The present study indicates that findings of previous studies regarding children’s proportional reasoning abilities should be carefully interpreted. Our study suggests that children’s abilities strongly differ depending on the proportional context that is used. These findings were obtained by directly comparing the role of context in one sample of children, using items that were as comparable as possible across contexts with respect to other potentially confounding variables. Using specific contexts, many researchers have concluded that young children cannot (e.g., Inhelder & Piaget, 1958; Noelting, 1980) or can reason proportionally from a young age (e.g., Boyer & Levine, 2012; Hurst & Cordes, 2018). Our research suggests that such conclusions need to be nuanced and that children can reason proportionally at an early age in some contexts, but not in others. In other words, context influences the level of difficulty and the mistakes children make. Following the same line of reasoning, we are aware that conclusions of our study should also be treated with caution. The present study only investigated children’s proportional reasoning abilities using missing-value tasks. Perhaps, differences between contexts regarding proportional reasoning are less prominent when other tasks are used, such as comparison tasks. Findings may also be different in an older or younger age group.

Moreover, in our study, children were presented to three different contexts in the same order. First, the probability context was presented, followed by the mixture context, and finally, the fair-sharing context. Based on previous research, we expected children to be proficient at fair-sharing. For the present study, it was important to grasp children’s spontaneous behavior in the different contexts. To avoid that children would be prompted to use fair-sharing strategies to tackle the probability and mixture contexts, the fair-sharing context was presented last. However, the set order of tasks could have still caused learning effects to occur, possibly affecting the results and leading to an overestimated proficiency in the fair-sharing context. Future research could investigate whether findings of the present study are replicable when contexts are presented in random order and when other types of task are being used. Moreover, including information about children’s decision-making and response times could strengthen our conclusions.

Future research could also investigate whether proportional reasoning in mixture and probabilistic missing-value contexts can be stimulated by teaching them to use fair-sharing schemes. For example, teaching children to distribute the winning elements over losing elements within a set in a probabilistic context might facilitate missing-value problems for children.

Our research also suggested that proportional problems might elicit different answering tendencies in children depending on the context in which the problem is presented. Typically, probabilistic contexts are seen as difficult for children to grasp, and probability is not or rather late included in the elementary school curricula. Although the probability context might be more difficult for children, resulting in a higher prevalence of typical erroneous answers of a certain sort, our research seems to suggest that there are also other contexts in which it is harder for children to reason proportionally (e.g., mixtures). Consequently, in interpreting research on children’s probabilistic reasoning abilities, we argue that one should be wary to avoid overgeneralizations of the findings to all proportional contexts.